Orbital Mechanics and Kepler's Laws

• Space dynamics, also known as orbital mechanics, is the study of the motion of artificial satellites and natural celestial bodies under the influence of gravitational and other forces
• It's a cornerstone of space mission design and operations

Kepler's Laws of Planetary Motion

• Kepler's First Law: Planets move in elliptical orbits with the sun at one focus
• Kepler's Second Law: A line joining a planet and the sun sweeps out equal areas during equal intervals of time
• Kepler's Third Law: The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit

Orbital Elements

• A set of six parameters that define a unique orbit
• Semi-major axis (a): Defines the size of the orbit, half of the longest diameter of the ellipse
• Eccentricity (e): Defines the shape of the orbit (0 for a circle, close to 1 for a highly elongated ellipse)
• Inclination (i): The angle between the orbital plane and a reference plane (e.g., the Earth's equator)
• Longitude of the ascending node (Ω): The angle between a reference direction (e.g., the vernal equinox) and the ascending node (where the orbit crosses the reference plane going north)
• Argument of periapsis (ω): The angle between the ascending node and the point of closest approach (periapsis)
• True anomaly (ν): The angle between the periapsis and the current position of the orbiting object, measured in the orbital plane

Coordinate Systems

• Essential for describing the position and velocity of objects in space
• Geocentric Equatorial Coordinate System: Uses the Earth's center as the origin and the equator as the reference plane
• Heliocentric Ecliptic Coordinate System: Uses the Sun as the origin and the ecliptic (the plane of Earth's orbit) as the reference plane

Two-Body Problem

• A simplified model that considers only the gravitational force between two point masses
• Provides a fundamental understanding of orbital motion
• Assumptions:
  • Only two bodies are present
  • Bodies are point masses
  • No external forces act on the system
• The solution to the two-body problem yields Kepler's laws

Orbital Maneuvers

• Change of orbit achieved through the application of thrust
• Hohmann Transfer: An elliptical orbit used to transfer between two circular orbits in the same plane
  • Fuel-efficient but time-consuming
• Bi-elliptic Transfer: Uses two half-ellipse burns to move between orbits
  • More fuel than Hohmann at some distances, less at other distances
• Inclination Change: Changing the angle of the orbit requires a large amount of energy

Perturbations

• Deviations from the ideal two-body motion due to additional forces
• Atmospheric Drag: A force that slows down satellites in low Earth orbit
  • Altitude decreases and orbital period shortens
• Gravitational Perturbations: Caused by the non-spherical shape of the Earth and the gravitational influence of other celestial bodies (Sun, Moon)
  • Can significantly alter orbital elements over time
• Solar Radiation Pressure: The force exerted by sunlight on a satellite, more significant for satellites with large surface areas

Lambert's Problem

• Determining the orbit that connects two positions in space at two given times
• Useful for trajectory design and rendezvous maneuvers
• Requires solving for the semi-major axis and other orbital elements

Rocket Propulsion

• Provides the thrust needed for orbital maneuvers and launches
• Specific Impulse (Isp): A measure of the efficiency of a rocket engine. Higher Isp means more thrust for the same amount of propellant
• Delta-v (Δv): A measure of the change in velocity required for a maneuver
  • Represents the performance of a rocket

Interplanetary Transfers

• Travel between planets
• Requires careful trajectory design to minimize fuel consumption and travel time
• Gravity Assist: Using the gravity of a planet to change a spacecraft's speed and direction
  • Saves fuel and shortens travel time

Orbital Stability

• The ability of an orbit to maintain its characteristics over time
• Affected by perturbations and gravitational forces
• Important for long-duration space missions

Spacecraft Attitude Dynamics

• The study of the orientation and rotation of spacecraft
• Necessary for pointing instruments, communication, and controlling maneuvers
• Attitude Control Systems: Use thrusters, reaction wheels, or control moment gyros to maintain desired orientation

Tensors in Space Dynamics

• Mathematical objects that describe physical quantities, such as stress or inertia, in a way that is independent of the coordinate system
• Useful for representing the inertia tensor of a spacecraft, which describes how the mass is distributed and affects its rotational motion

Chaos Theory in Space Dynamics

• Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions
• Chaotic orbits can arise in multi-body systems or due to complex gravitational fields
• Important for understanding the long-term stability of orbits and designing missions in chaotic environments

Atmospheric Entry

• The process of a spacecraft entering a planet's atmosphere
• Generates significant heat due to friction with the atmosphere
• Requires thermal protection systems to prevent spacecraft from burning up

Relative Motion

• The motion of one object relative to another
• Important for rendezvous and docking maneuvers, formation flying, and close proximity operations
• Clohessy-Wiltshire equations: a set of linear equations describing the relative motion of two spacecraft in close proximity

Space Debris

• Inactive satellites, rocket bodies, and fragments orbiting Earth
• Poses a collision risk to operational spacecraft
• Mitigation strategies include deorbiting spacecraft at the end of their lives and actively removing debris from orbit